Study if the following functions are increasing / decreasing at point $$x=0$$.
a) $$y=x^3$$
b) $$y= \left\{ \begin{array} {rcl} 0 & \mbox{ if } & x \leq 0 \\ -x & \mbox{ if } & x>0 \end{array}\right.$$
Development:
a) See the graph
At a first glance we see that the graph is an increasing function, although in $$x=0$$ it is less clear what is happening. Is it, then, a strictly increasing function in $$x=0$$?
Let's calculate it analytically. For that let's calculate the derivative: $$$y'=3x^2$$$ Let's see what the sign of the derivative is in the points placed in a environment to $$x=0$$.
We can see that for any value of $$x$$ (different from zero) the derivative is positive. Therefore all the points of the environment of $$x=0$$ have positive derivative. This means that the function is strictly increasing in $$x=0$$.
b) See the derivative in values close to $$x=0$$.
For negative values of $$x$$, the derivative $$y'=0$$.
For positive values of $$x$$, the derivative $$y'=-1$$.
Therefore, $$y'\leq0$$ in an environment to $$x=0$$, and therefore the function is decreasing at $$x=0$$ (it is not strictly decreasing!)
Solution:
a) Strictly increasing
b) Decreasing